Date: Dec 8, 2012 3:57 AM Author: Zaljohar@gmail.com Subject: Re: Background Theory On Dec 7, 7:21 pm, fom <fomJ...@nyms.net> wrote:

> On 12/7/2012 8:46 AM, Zuhair wrote:

>

> > One might wonder if it is easier to see matters in the opposite way

> > round, i.e. interpret the above theory in set theory? the answer is

> > yes it can be done but it is not the easier direction, nor does it

> > have the same natural flavor of the above,

> > it is just a technical formal piece of work having no natural

> > motivation. Thus I can say with confidence that the case is that Set

> > Theory is conceptually reducible to Representation Mereology and not

> > the converse!

>

> I have no doubt that you are correct. In another post

> in your thread I summarized the work of Lesniewski which

> uses the part relation to characterize classes. His

> method was specifically designed to circumvent the

> grammatical form that leads to Russell's paradox.

>

> Moreover, the historical record of the philosophy sides

> with you. Aristotle, Leibniz, Kant, and Russell all

> have had positions asserting how parts are prior to

> individuals.

>

> The issue becomes, however, whether the theory informs

> differently. I turned to the investigation of parts

> without any knowledge of mereology because I believe

> that the ontological interpretation of the theory of

> identity is inappropriate for foundational purposes.

> Find any distinction in the literature between the

> application of logic to a linguistic analysis and

> the linguistic synthesis of a theory.

>

> That is why my proper part relation has a the

> character of a self-defining predicate. There must

> be a first asserted truth, and, it must be an

> assertion that constructs the language.

>

> The second sentence has parallel syntax to the

> first. It took me a long time to understand

> why I should accept that situation. In Liebniz'

> logical papers, one finds the remark that individuals

> should be identified with a mark. The membership

> relation is precisely that relation which marks an

> individuated context. It does so in relation to

> a plural context.

>

> The priority of the part relation as it applies

> to set theory is that the fundamental relation

> between objects of a domain should reflect the

> object type of the domain. To the extent that

> sets are

>

> "collections taken as an object"

>

> the fundamental relation should relate collections

> to collections. To the extent that a set is

>

> "determined by its elements"

>

> requires that the individuated context be situated

> relative to the defining syntax of the fundamental

> relation.

>

> One of the primary focuses of my investigation

> was to understand the principle of identity of

> indiscernibles. If you read Leibniz, you will

> find that what is taught about the identity of

> indiscernibles is not entirely representative

> of what Leibniz said.

>

> He attributes his motivation to a position held

> by Thomas Aquinas and explains that the typical

> formulation is a consequence of the phrase

>

> "an individual is the lowest species"

>

> This is best interpreted mathematically by

> Cantor's nested closed set theorem with vanishing

> set diameter.

>

> The following assumption:

>

> Assumption of Aquinian individuation:

> AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) ->

> Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw)))))

>

> can be used to define membership

>

> AxAy(xey <-> (Az(ycz -> xez) /\

> Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw))))))

>

> relative to a prior proper part relation "c". In both

> sentences, the complexity arises from capturing the sense

> of a nested sequence of sets.

>

> To be quite honest, here, I still need to consider

> these expressions further. These universals are

> difficult because both the assertion and its

> negation must be considered.

>

> So, let me suggest that while you continue

> to refine your ideas, you also develop some of

> the claims you have made concerning

> the representations of arithmetic and such.

>

> Do not take these remarks wrong. I see that you

> are reading other authors and developing your

> ideas. So, please continue. I would like to

> see more.

>

> Inform us.

I just wanted to note that with this approach a set is a singular

entity that

represent a plurality of singular entities, while with Lewis's

approach

a set is a plurality of singular entities that is represented by a

singular

entity. I see this approach reductive while Lewis's diffusive. However

formally speaking they almost mirror each other, but conceptual wise

I think the approach given here is more faithful to the general

context

in which sets are mentioned.

Zuhair